In a 200-m linear race, if A gives B a start of 25 m, then A wins the race by 10 seconds. Alternatively, if A gives B a start of 45 m, the race ends in a dead heat. How long does A take to run 200m?

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  1. 78 seconds
  2. 77 seconds
  3. 78.5 seconds
  4. 77.5 seconds

Answer (Detailed Solution Below)

Option 4 : 77.5 seconds
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Detailed Solution

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Given

In a 200-m linear race, if A gives B a start of 25 m, then A wins the race by 10 seconds.

Formula used

Distance = speed × time

Calculation 

Let’s assume speed of A as va and speed of B as vb

Then, in a 200 M race distance run by A = 200 m

Distance run by B after getting head start of 25 M = 175 m

Time taken by A = 200/va

Time taken by B = 175/vb

Since B lacks behind by 10 sec therefore Time taken by B is more as compare to that of A by 10 sec,

⇒ 175/vb  - 200/va  = 10 ....(1)

if A gives B a start of 45 m, the race ends in a dead heat,

Time taken by B = (200 - 45)/vb = 155/vb

Time taken by A = 200/va

Both finish at same time,

∴ 155/vb = 200/va

⇒ va = 40/31 × vb       ----(2)

Putting (2) in (1) we get,

Vb = 2 

Time taken by B to complete 200M race would be 175 / 2 = 87.5 seconds

Thus time taken by A would be = 87.5 - 10 = 77.5 second

A takes 77.5 seconds to run 200m.

Alternate Method 

In the first case:

A runs = 200m , B runs = (200-25)m = 175m

Let A take x s to run 200m then, B will take (x+10) s for 175m 

In the second case:

A runs = 200m, B runs = (200-45)m = 155m

Time taken by B to run 155m = time taken by A to run 200m = x s

From the first and second cases, it is clear that for (175-155)m = 20m extra run B takes 10 s extra time.

Thus speed of B = 20m/10s = 2m/s

Therefore, Time taken by A to run 200m = Time taken by B to run 155m = 155/2 = 77.5 s

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